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Let K be a planar convex body with C^1 boundary. Consider a random line G intersecting K. In [1] Pleijel discovered an identity expressing the integral functional of the length of the chord G ∩ K in terms of the double integration over the boundary of K. From this identity it is possible to derive an explicit form of the defect in the isoperimetric inequality and show that it is nonnegative. There exists ([2]) an analogue of the Pleijel identity for the convex bodies with smooth boundary in R^3 . In [2] Ambartzumian presented version of Pleijel identity for the convex planar polygons which is now known as the Ambartzumian–Pleijel identity. We present generalizations of the Pleijel and Ambartzumian–Pleijel identities to arbitrary dimension. Moreover, we present the generalisation of Blaschke– Petkantschin ([3], Theorem 7.2.7) and Zahle ([4]) formulae for convex bodies ¨ with smooth boundary.
[1] Pleijel, A. Zwei kurze Beweise der isoperimetrischen Ungleichung. // Archiv der Mathematik, 7(4): 317–319, 1956.
[2] Ambartzumian, R.V. Combinatorial integral geometry: with applications to mathematical stereology. – John Wiley & Sons – 1982.
[3] Schneider, R. and Weil, W. Stochastic and integral geometry. – Probability and its Applications (New York). Springer-Verlag, Berlin – 2008.
[4] Zahle, M. A kinematic formula and moment measures of random sets. // Mathematische ¨ Nachrichten, 149(1): 325–430, 1990. | 
